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First-passage-time densities and avoiding probabilities for birth-and-death processes with symmetric sample paths

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Antonio Di Crescenzo
Antonio Di Crescenzo*
Affiliation:
Università di Napoli ‘Federico II’
*
∗Postal address: Dipartimento di Matematica e Applicazioni ‘R. Caccioppoli’, Università di Napoli ‘Federico II’, via Cintia, 80126, Napoli, Italy. E-mail address: dicrescenzo@matna1.dma.unina.it
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Abstract

For truncated birth-and-death processes with two absorbing or two reflecting boundaries, necessary and sufficient conditions on the transition rates are given such that the transition probabilities satisfy a suitable spatial symmetry relation. This allows one to obtain simple expressions for first-passage-time densities and for certain avoiding transition probabilities. An application to an M/M/1 queueing system with two finite sequential queueing rooms of equal sizes is finally provided.

Keywords

Truncated processestransition probabilitiesspatial symmetryabsorptionreflectionfirst-passage timeavoiding transition probabilitiesM/M/1 queueing systembusy period

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 2 , June 1998 , pp. 383 - 394
Copyright
Copyright © Applied Probability Trust 1998 

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Footnotes

Work supported in part by Italian National Research Council (CNR) under Contracts Nos. 95.742.01, 95.1090.01, 96.180.01, 96.3859.01 and by MURST (40% funds).

References

Abate, J., Kijima, M., and Whitt, W. (1991). Decompositions of the M/M/1 transition function. Queueing Systems 9, 323–336.CrossRefGoogle Scholar
Abramowitz, M., and Stegun, I.A. (1972). Handbook of Mathematical Functions. Dover, New York.Google Scholar
Anderson, W.J. (1991). Continuous-Time Markov Chains. Springer, New York.CrossRefGoogle Scholar
Böhm, W., and Mohanty, S.G. (1994). On random walks with barriers and their application to queues. Studia Sci. Math. Hung. 29, 397–413.Google Scholar
van Doorn, E.A. (1981). On the time dependent behaviour of the truncated birth-death process. Stoch. Proc. Appl. 11, 261–271.CrossRefGoogle Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. I, 3rd edn. Wiley, New York.Google Scholar
Giorno, V., Negri, C., and Nobile, A.G. (1985). A solvable model for a finite-capacity queueing system. J. Appl. Prob. 22, 903–911.CrossRefGoogle Scholar
Giorno, V., Nobile, A.G., and Ricciardi, L.M. (1989). A symmetry-based constructive approach to probability densities for one-dimensional diffusion processes. J. Appl. Prob. 26, 707–721.CrossRefGoogle Scholar
Karlin, S. (1964). Total positivity, absorption probabilities and applications. Trans. Amer. Math. Soc. 111, 33–107.CrossRefGoogle Scholar
Karlin, S., and Mc Gregor, J.L. (1957). The differential equations of birth-and-death processes, and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85, 489–546.CrossRefGoogle Scholar
Karlin, S., and Mc Gregor, J.L. (1957). The classification of birth and death processes. Trans. Amer. Math. Soc. 86, 366–400.CrossRefGoogle Scholar
Keilson, J. (1971). Log-concavity and log-convexity in passage time densities of diffusion and birth-death processes. J. Appl. Prob. 8, 391–398.CrossRefGoogle Scholar
Keilson, J. (1979). Markov Chain Models–-Rarity and Exponentiality. Applied Mathematical Science Series 28. Springer, Berlin.CrossRefGoogle Scholar
Keilson, J. (1981). On the unimodality of passage time densities in birth-death processes. Statist. Neerlandica 25, 49–55.CrossRefGoogle Scholar
Kijima, M. (1988). On passage and conditional passage times for Markov chains in continuous time. J. Appl. Prob. 25, 279–290.CrossRefGoogle Scholar
Lánský, P., and Rospars, J.-P. (1993). Coding of odor intensity. BioSystems 31, 15–38.CrossRefGoogle ScholarPubMed
Mohanty, S.G., and Panny, W. (1990). A discrete-time analogue of the M/M/1 queue and the transient solution: a geometric approach. Sankhya Ser. A. 52, 364–370.Google Scholar
Ricciardi, L.M. (1986). Stochastic population theory: birth and death processes. In Mathematical Ecology, eds Hallam, T.G. and Levin, S.A. (Biomathematics 17.) Springer, pp. 155–190.CrossRefGoogle Scholar
Roehner, B., and Valent, G. (1982). Solving the birth and death processes with quadratic asymptotically symmetric transition rates. SIAM J. Appl. Math. 42, 1020–1046.CrossRefGoogle Scholar
Rösler, U. (1980). Unimodality of passage time density for one-dimensional strong Markov processes. Ann. Prob. 8, 853–859.CrossRefGoogle Scholar
Sumita, U., and Masuda, Y. (1987). Classes of probability density functions having Laplace transforms with negative zeroes and poles. Adv. Appl. Prob. 19, 632–651.CrossRefGoogle Scholar
Xie, S., and Knessl, C. (1993). On the transient behavior of the M/M/1 and M/M/1/K queues. Studies Appl. Math. 88, 191–240.CrossRefGoogle Scholar